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Abstract

Background

In the absence of stimuli, most motile eukaryotic cells move by
spontaneously coordinating cell deformation with cell movement in the
absence of stimuli. Yet little is known about how cells change their own
shape and how cells coordinate the deformation and movement. Here, we
investigated the mechanism of spontaneous cell migration by using
computational analyses.

Methodology

We observed spontaneously migrating Dictyostelium cells
in both a vegetative state (round cell shape and slow motion) and
starved one (elongated cell shape and fast motion). We then extracted
regular patterns of morphological dynamics and the pattern-dependent
systematic coordination with filamentous actin (F-actin) and cell
movement by statistical dynamic analyses.

Conclusions/Significance

We found that Dictyostelium cells in both vegetative and
starved states commonly organize their own shape into three ordered
patterns, elongation, rotation, and oscillation, in the absence of
external stimuli. Further, cells inactivated for PI3-kinase (PI3K)
and/or PTEN did not show ordered patterns due to the lack of spatial
control in pseudopodial formation in both the vegetative and starved
states. We also found that spontaneous polarization was achieved in
starved cells by asymmetric localization of PTEN and F-actin. This
breaking of the symmetry of protein localization maintained the leading
edge and considerably enhanced the persistence of directed migration,
and overall random exploration was ensured by switching among the
different ordered patterns. Our findings suggest that
Dictyostelium cells spontaneously create the
ordered patterns of cell shape mediated by PI3K/PTEN/F-actin and control
the direction of cell movement by coordination with these patterns even
in the absence of external stimuli.

Introduction

Cell migration, which is a fundamental cellular response in inflammatory
reactions and other physiological activities, is a highly complex process that
integrates many spatial and temporal cellular events [1]–[6]. Motile
bacteria and most eukaryotic cells can move in a directed (directed cell
migration) or spontaneous (spontaneous cell migration) fashion depending on the
presence or absence of external cues.

Directed cell migration toward a soluble ligand or chemotaxis is a general
property of many motile eukaryotic cells [7].
Dictyostelium cells extend one or more pseudopodia at a
time, with the extended pseudopodia being close to the chemoattractant source.
The sensing of chemotactic cues is achieved by the activation of
G-protein-coupled receptors and the cytosolic regulator of adenylyl cyclase
(CRAC) [8]. To polarize and migrate up the
chemotactic gradient after sensing chemical cues, cells restrict filamentous
actin (F-actin) polymerization and form a leading edge. This task requires the
cells to amplify a shallow external chemical gradient into a steep intracellular
response. A recent study has revealed that, during chemotaxis in
Dictyostelium cells, phosphoinositide 3-kinase (PI3K) is
localized at the leading edge of the cells and new pseudopodia [9]. A counteracting phosphatase of PI3K, the
phosphatase and tensin homolog, (PTEN), is simultaneously localized at the rear
edge of the cell [10]. This reciprocal localization of the
two proteins creates a spatial internal gradient of phosphatidyl inositol
tri-phosphate (PIP3) across the plasma membrane; the level of
PIP3 is higher at the leading edge than at the rear edge due to
the kinase activity of PI3K.

On the other hand, motile cells are able to migrate spontaneously even in the
absence of external stimuli. In the absence of external cAMP, individual
Dictyostelium cells spontaneously form actin filaments and
extend 1 or 2 pseudopodia. The new pseudopodia are retracted or attached to the
substrate. After the attachment of the pseudopodia to the substrate, the cell
adopts a polarized morphology for a few minutes and then retracts its rear edge
and moves forward. Spontaneous cell migration allows the cells to forage and
explore their surroundings by balancing random and directed migrations. It is
thought that Dictyostelium cells extend pseudopodia in a more
or less random manner [7]. Recently,
Dictyostelium cells were observed to undergo spontaneous
migration through stochastic activation of both PI3K and Ras at the sites of new
pseudopod formation even in the absence of either chemoattractants or functional
heterotrimeric G-proteins [11], [12]. However, the
fundamental question of how a cell demonstrates spontaneous migration by
randomly remodeling its shape through underlying molecular interactions of
PI3K/PTEN/F-actin remains unanswered. Here, in order to reveal the mechanism of
spontaneous cell migration, we employed a quantitative approach based on the
statistical analysis of the cell shape of single Dictyostelium
cells. To gain a further understanding of spontaneous cell migration, we
observed two different developmental states of Dictyostelium
cells, namely the vegetative and starved states. Dictyostelium
cells exhibit vegetative and starved states at different developmental stages,
but they are able to move spontaneously in both of states. Vegetative cells are
more round in shape and move more slowly than starved cells; comparing these two
cell states enables us to identify common mechanisms underlying cell migration.

We analyzed the stochastic dynamics of cell shape and their coordination with
cell movement by using correlation analyses such as autocorrelation functions.
Correlation analysis is a useful method for identifying repeating signal
patterns that are masked by noise. We observed that despite apparently random
pseudopodia formation, the remodeling of the cell shape was organized into three
distinct ordered patterns: elongation, rotation, and oscillation. The ordered
remodeling of cell shape was correlated with PI3K-dependent F-actin
polymerization. PI3K activity was required for the formation of actin-filled
pseudopodia while PTEN restricted the formation of excess pseudopodia; this
supports the idea that the ordered patterns are mediated by PI3K/PTEN/F-actin.
Furthermore, we observed that F-actin and PTEN spontaneously localize at the
leading and rear edge of the cell, respectively. The asymmetric localization of
both proteins would ensure the reinforcement of the leading edge, which would
facilitate the maintenance of a biased direction of cell migration. Eventually,
the cells would be able explore their surroundings by switching among the
ordered patterns. Our results suggest that the ordered patterns of cell shape
mediated by PI3K/PTEN/F-actin result in spontaneous cell migration even in the
absence of chemotactic cues.

Results

Ordered remodeling of cell shape in spontaneously migrating cells

We observed single Dictyostelium cells in the absence of
chemoattractants or nutrients at a low cell density. We made time-lapse
movies over 500–600 s, at a resolution of 1 frame per second.
Dictyostelium cells displayed two phases, a vegetative
(VEG) state and a starved (STA) state (Fig. 1A). WT STA cells were highly motile
and polarized with a contracted tail and a more distinct anterior lamella
than WT VEG cells. These cells formed well-defined pseudopodia and crawled
in an apparently random direction. The morphological dynamics of cell shape
appeared to be random in both states, and initially it was difficult to
discern any rules governing the changes in cell shape (Movies
S1–S6).

First, we constructed a circular map around the centroid of a cell in order
to describe the dynamic remodeling of cell shape (Fig. 1B and see Methods). We defined the variable
Amp(θ,t) as the distance
from the centroid to the edge of the cell membrane in the direction
θ and at time t.
θ = 0°
was fixed in the right-hand direction on the horizontal axis. Figure 2 shows the
Amp(θ,t) of both WT STA
and WT VEG cells. Figure
2A shows that WT STA cells exhibit various dynamics of cell shape
during spontaneous cell migration. WT STA cell 1 maintained a larger
Amp(θ,t) at both
80° and 260° angles, meaning that the cell stably
maintained its elongated morphology (Fig. 2A). However, it is often difficult
to recognize patterns in
Amp(θ,t) because of the
intrinsic noise of cell deformations. In order to identify repeating
patterns governing morphological dynamics, we calculated the
auto-correlation function (ACF) of
Amp(θ,t) (see Methods) [13].
Remarkably, we found ordered patterns (Fig. 2A, lower row). WT STA cell 1
exhibited a stable positive correlation along the lines
Δθ = 0°
and
Δθ = ±180°,
indicating that this cell maintains an elongated shape (elongation pattern).
In WT STA cell 2, the correlation was laterally propagating, indicating the
lateral propagation of pseudopodia (rotation pattern). WT STA cell 3
exhibited an oscillatory periodic profile with slight rotation; it reshaped
its membrane periodically by extending pseudopodia in a certain direction
and then re-extending at certain angles to the long axis of the cell in
cycles of 2.5 min (oscillation pattern). Moreover, WT STA cell 1 exhibited
more noisy Amp(θ,t) at the
leading edge (Amp(260,t)) rather than at
the rear edge (Amp(80,t)), indicating that
lamellipodia are formed predominantly at the leading edge. The branches
propagating from the leading edge to the rear imply that some of the
lamellipodia adhere to the substrate and become fixed while the whole cell
body moves forward (Fig.
2A, arrowheads). We also note that cells exhibiting a rotation or an
oscillation pattern show variability of from 2.5 min to 3.5 min in the
period of rotation and oscillation. We identified ordered patterns in nearly
70% of WT STA cells, and transient patterns that were
intermediate between the two ordered patterns in the remaining cells
(n = 53). There was no significant
difference in the average area of cells exhibiting the different patterns
(p>0.05, ANOVA). Our findings further suggest that WT STA cells
produce three ordered patterns without the need for external stimuli.

Ordered patterns of cell shape are common in both vegetative and starved
cells

We next examined the morphological dynamics of WT VEG cells. WT VEG cells
were less motile than WT STA cells, and their shapes are less polarized
(Table 1). To our
surprise, despite the significant decrease in migration speed and in the
roundness of the cells, WT VEG cells also exhibited ordered patterns similar
to those of the WT STA cells (Fig. 2B); WT VEG cells 1, 2, and 3 were deformed by elongation,
rotation, and oscillation, respectively. When a cell extends pseudopodia in
a certain direction and then re-extends new pseudopodia perpendicular to the
long axis of the cell, an oscillation pattern occurs. 70% of the
WT VEG cells exhibited the ordered patterns, and the remaining cells
exhibited transient patterns that were intermediate between the two ordered
patterns (n = 53). The area of a cell
does not appear to determine the type of pattern since the average areas for
WT VEG cells of each of the three pattern types were not significantly
different (p>0.05, ANOVA). These three commonly observed
patterns indicate that the morphological dynamics of WT VEG cells are also
organized into ordered patterns without the need for external stimuli. The
rates of occurrence of each pattern were roughly the same between vegetative
and starved cells although the oscillation pattern was less frequently
observed than the than other two patterns. A previous study by Killich et
al. has identified rotation and oscillation patterns in WT STA cells using a
different type of analysis [2]. These patterns clearly corresponded
to the rotation and oscillation patterns that we observed. Moreover, our
comprehensive analysis revealed that WT STA cells also show an elongation
pattern and that three types of ordered pattern are commonly observed in
vegetative states. Our results not only validate the previous observation
but also fully characterize the dynamics of cell shape in both STA and VEG
states.

The fact that certain patterns were commonly observed in the different cell
types raise a question: whether the transition of ordered patterns occurs or
not in a single cell. To answer this question, we performed long-term
observation of single WT cells for more than 30 min. We then calculated the
ACF of long-term Amp(θ,t) data
by averaging over moving windows of 10 min. Figure 3A shows that the WT STA cell
exhibited the elongation pattern until 10 min, then switched to the 72
degree/min rotation pattern. Figure 3B shows that the WT VEG cell also exhibited the
transition between ordered patterns, but in this case as a change from a
right-handed rotation pattern of 36 degree/min (2 to 20 min) to a
left-handed rotation pattern of 36 degree/min (20 to 30 min). Moreover, 3.3
hour measurement of a WT vegetative cell revealed that a single cell can
show three types of pattern, a rotation, elongation, and oscillation pattern
(Fig.
S1). However, when we calculated the ACF of this cell by averaging
the Amp(θ,t) data over 3.3
hour, we no longer observed ordered patterns (data not shown). These results
indicate that the cells showed ordered patterns lasting for 10 to 20 min and
that the patterns then changed spontaneously via the stochastic transition
of the patterns.

Ordered patterns of cell shape are mediated by PI3K and PTEN

We demonstrated that morphological dynamics are spontaneously organized into
three ordered patterns. What kind of molecule(s) controls these patterns? In
order to address this question, we explored the specific molecular events
that affect the patterns. We focused on two molecules, PTEN and PI3K. PTEN
is a lipid phosphatase that dephosphorylates
phosphatidylinositol-3,4,5-trphosphate (PIP3) to
phosphatidylinositol-4,5-biphosphate (PIP2), whereas PI3K is a
counteracting kinase that reciprocally converts PIP2 to
PIP3. Reciprocal regulation of PIP3 by PI3K and PTEN
involves in maintaining cell polarity and in efficient chemotaxis [7], [14]–[20]. The PI3K
pathway is also linked to the control of F-actin polymerization mediated by
RacB activation and the RacGEF1 translocation to the plasma membrane [21]–[23]. The
PI3K-dependent F-actin polymerization is essential for reinforcing the
leading edge during chemotaxis. However, apart from their association with
cell polarity and chemotaxis, how the PI3K pathway acts cooperatively to
control both the dynamics of cell shape and its coordination with
spontaneous migration is poorly understood.

We initially examined the morphological dynamics of the pten
null mutant (pten−).
pten− cells migrated more slowly
than WT cells in both the VEG and STA states, and exhibited a less polarized
shape with broad pseudopodia (Table 1, Fig. S2, Movies
S7 and S8). We found much less ordered patterns
in pten− cells in either the VEG or
STA states compared with WT cells
(n = 66 in VEG and
n = 53 in STA; Figs 4A and 4B).
pten− cells were defective in
the spatially restricted formation of pseudopodia and failed to organize the
patterns, probably because the absence of 3-phosphatase activity by PTEN
leads to an increase in the proportion of the membrane area containing
proteins with a PH domain, and rapid and erratic expansion of the region
from which actin-filled pseudopodia are extended [7],
[10], [16], [24]. Thus, PTEN is a key regulator for
organizing morphological dynamics by suppressing excess pseudopodia in both
the VEG and STA states. We therefore investigated the morphological dynamics
of WT cells treated with the specific inhibitor LY294002 (PI3K-inhibited
cells) (n = 16 in VEG and
n = 22 in STA) and the
pi3k1/2 null mutant
(n = 20 in VEG and
n = 18 in STA). The PI3K-inhibited cells
exhibited an aberrant cell shape and were poorly polarized (Movies
S9–S12), and rarely displayed ordered
patterns in both the VEG and STA states (Fig. 4C); 5% of
PI3K-inhibited cells exhibited a rotating pattern, which may have been due
to either the incomplete inhibition of PI3K activity or to redundant
pi3k genes. In order to further clarify the effect of
PI3K and PTEN further, we examined the morphological dynamics of the double
mutant, pten−+LY294002
cells. This mutant also exhibited a less polarized cell shape with a single
(or sometimes multiple) pseudopodium and less motility (Movies
S13 and S14). We did not observe ordered patterns
in the double mutant, suggesting that the timing and the direction of
pseudopodia became to be random due to the inhibition of both PI3K and PTEN
(n = 30 in VEG and
n = 33 in STA; Fig. 4D). Hence, PI3K and PTEN but not
external signals are required to induce ordered remodeling of the cell shape
from random morphological dynamics in both the developmental stages.

Ordered patterns of morphological dynamics are disorganized
following knockdown of PTEN and following inhibition of PI3K.

PI3K-dependent actin polymerization correlates with ordered patterns

We next considered the specialized role of PI3K and PTEN in the control of
cell shape. In order to gain insight into the PI3K-dependent organization of
morphological dynamics at the molecular level, we investigated the dynamics
of PI3K-dependent F-actin polymerization in both WT VEG and WT STA cells.
Measurement of actin binding domain fused GFP (ABD-GFP) expression was used
as an index of F-actin assembly and localization [25] (Fig. 5A). We measured the
dynamics of F-actin accumulation
(Act(θ,t)) and the
dynamics of cell shape
(Amp(θ,t)) every 6 s, and
examined the correlation between these processes using a cross-correlation
function (CCF; see Methods). We found
that WT cells showed a large correlation between
Act(θ,t) and
Amp(θ,t) in both the VEG
and STA states, suggesting that F-actin was predominantly accumulated in the
elongated region of the cell (Fig. 5B). In contrast, PI3K-inhibited cells exhibited
fast-decaying accumulation of F-actin but no significant correlation between
Act(θ,t) and
Amp(θ,t) in both the VEG
and STA states (Fig.
5C). The high degree of cross-correlation suggests that PI3K activity
regulates the amplitude of pseudopodia through PI3K-dependent actin
polymerization.

In addition to the CCF between
Amp(θ,t) and
Act(θ,t), we further
examined the specialized role of PI3K and PTEN in the control of cell shape
by using power spectrum of
Amp(θ,t). The power spectrum
of Amp(θ,t) provides
additional information than the CCF on the mechanism underlying the
development of ordered patterns because it is a measure of the amplitude of
pseudopodia at scales of different length. We found that the spectrum of
PI3K-inhibited cells had lower amplitude, suggesting that PI3K-inhibited
cells rarely grow large pseudopodia (Fig. S3). Moreover, a small spectrum
amplitude is a commonly observed property in both the VEG and STA states.
These results are consistent with the strong correlation of the CCF in WT
cells (Fig. 5). In
addition, the amplitude of the spectrum of
pten− cells was comparable to
that of the WT cells, suggesting that PTEN disruption did not affect the
amplitude of pseudopodia (Figs. S2 and S3). It
appears that PTEN does not control the amplitude of pseudopodia but is
essential for the formation of spatially restricted pseudopodia as a
suppressor of random pseudopodial activity. Furthermore, treatment of
pten− cells with LY294002 also
resulted in a spectrum of smaller amplitude. These results support our
conclusion in the preceding section: the initiation of an ordered remodeling
of cell shape from random morphological dynamics is mediated by both the
PI3K-dependent actin polymerization and the suppression of excess
pseudopodia by PTEN.

Coordination between ordered remodeling of cell shape and cell movement

Finally, we addressed a central problem of cell migration, the relationship
between morphological dynamics and cell migration. We defined the
persistence length of centroid motion as a characteristic length of
persistence of directional migration (see Methods).. The persistence of directional migration of WT STA
cells gradually decays as a function of distance, while that of both
pten− STA and
pi3k1/2− STA cells, and also of
WT VEG cells, rapidly reaches to 0 (Figs. 6A and S4,
Text
S1); The averaged persistence length of WT STA cells (16.4
µm) was 27 times greater than those of both
pten− STA and
pi3k1/2− STA cells, and also
that of WT VEG cells (Table
1). The difference of persistence length is not explained solely by
the difference of velocity because WT STA cells move four times faster than
WT VEG cells. In order to reveal the mechanism by which WT STA cells create
persistent motion in the absence of external signals, we investigated the
coordination between cell shape and cell movement. We calculated the
probability distribution of the deviation of
Amp(θ,t) in the direction
of migration, (see Methods). This
distribution tells us whether a cell tends to extend (or retract) its
membrane in the direction of migration. The probability distribution of WT
STA cells was largely biased towards the positive side (), whereas that of both
pten− STA and
pi3k1/2− STA cells, and also of
WT VEG cells, was only slightly shifted toward the positive side with (Fig.
6B). The bias of WT STA cells was approximately 6 times larger than
that of the other cell. The difference in distribution profiles confirms
that WT STA cells directly migrate in the direction of protrusions without
the necessity of external cues, while PTEN disruption or PI3K1/2 deletions
diminished such coordination between cell polarity and cell migration.
Furthermore, we observed the spatial localization of PTEN-GFP and ABD-GFP
and found that PTEN and F-actin were spontaneously localized at the rear and
the leading edges asymmetrically in WT STA cells, respectively [12] (Figs. 6C and 6D). We did not observe the
asymmetric localization of PTEN-GFP at the membrane in WT VEG cells. In
addition, the localizations of these molecules were abolished by inhibition
of actin polymerization either by latrunculinA or by LY294002 (data not
shown), indicating that actin polymerization is required for establishing
asymmetric localization of PTEN. Our observations imply that the breaking of
spontaneous symmetry accompanied by localization of PTEN reinforces the cell
polarity and that the marked change in persistence length between the VEG
and STA states is controlled through the persistent asymmetric localization
of F-actin mediated by the PI3K pathway. Furthermore, we evaluated the
correlation between the membrane extensions (or retractions) along the
direction of cell movement,
Amp(θ,t) at time
t = t+Δt
and the centroid displacements V(t) at
time t = t by using the
identical angle cross-correlation function (iaCF) as follows (see Methods); the iaCF is useful for
evaluating the coordination between cell deformation and cell movement for
individual cells. When we obtain the maximum value of the iaCF at
Δt = τ,
this cell tends to protrude its own membrane toward the direction of cell
movement with a time delay of τ. We found four
times larger positive correlation in the WT STA compared with the WT VEG
cells, pten− STA cells, and
pi3k1/2− STA cells at
Δt = 0
(Fig. 6E). This
result indicates that the WT STA cells coordinate the extension and
retraction of protrusions with cell movement and that the PI3K inhibition
and pten depletion causes the loss of such coordination.

We further investigated the correlation between the ordered pattern and cell
movement by examining the persistence length on each pattern in the WT STA
cells. The average persistence length of elongating WT STA cells (25
µm) was larger than that of rotating or oscillating cells (9
µm) (Fig.
7A). The dependency of the persistence on the type of ordered
patterns suggests that key proteins related to the pattern regulate the
location of F-actin accumulation and then steer cell movement. Moreover, we
examined this pattern-oriented migration again by comparing iaCFs between
the elongation pattern and rotation/oscillation pattern. The iaCF of the
elongation pattern showed a long-lasting positive correlation within
−300
sec<Δt<300 sec, while
the iaCF of the rotation/oscillation pattern rapidly decayed and eventually
reached to 0 near
Δt = ±100
sec (Fig. 7B). This
result means that the elongating cells maintain the elongated cell shape in
the direction of motion, and rotating or oscillating cells rapidly turn in
accord with the deformation of cell shape. Thus, the ordered patterns of
cell shape are coordinated with cell movement even in the absence of
external stimuli. The coordination was observed solely in WT STA cells, and,
as a separate observation, the asymmetric localization of PTEN occurred in
WT STA cells. This suggests that spontaneous cell polarization assisted with
the reciprocal localization of PI3K and PTEN, which in turn was responsible
for the strong correlation between the patterns and cell migration.

Discussion

In this study, we demonstrated that the ordered patterns of cell shape, which are
dissected into three patterns, are observed in both vegetative and starved
cells. A previous study reported rotation and oscillation patterns in starved
cells [2]; however, the elongation pattern in WT
STA and the ordered patterns in WT VEG cells have not been reported. Because
there have been few studies investigating the morphological dynamics of
vegetative cells, whether these three ordered patterns are common to different
developmental stages or specific only to starved Dictyostelium
cells remains to be determined. We here demonstrated that these three patterns
are also observed in VEG cells, meaning that the timing and direction of
pseudopod extension-retraction is not random but organized into ordered patterns
even in VEG cells. Our result encourages us to reexamine the conventional notion
of spontaneous membrane dynamics in Dictyostelium cells, which
is that in the absence of a chemoattractant gradient, unpolarized
Dictyostelium cells extend pseudopodia in more or less
random directions.

Recently, Li et al. examined the persistent cell locomotion of
Dictyostelium cells using long term observations and
statistical analysis of centroid locations [26]. They identified
that the amoeboid motion could be decomposed into the several characteristic
motions: the circular motion of approximately 36 degree/min (CIRC), the
oscillation motion with period of 2 to 3 min (OSC), and the instantaneous large
turn. We could not identify the instantaneous large turn in our data, but found
the cells moving in a fashion similar to CIRC and OSC: The cells in the rotation
pattern of cell shape moved in a fashion with angular speed of 36 to 45
degree/min (n = 13; Fig. S5).
Moreover, the duration of the rotation pattern was 10 to 20 min that was
comparable to that of CIRC. In addition, the oscillation pattern of cell shape
showed the period of 2.5 to 3.5 min that was approximately equivalent to that of
OSC (2 to 3 min). According to these comparisons, we suppose that the
characteristic motions of CIRC and OSC were attributed to the coordination of
cell movement with the rotation and oscillation patterns of cell shape,
respectively.

We mention that OSC showed the various durations from minutes to an hour though
the oscillation pattern of cell shape was maintained for 10 to 20 min but not an
hour. We consider that this is because the number of our measurements was not
enough to find the oscillation pattern of an hour.

Although the mechanism by which PTEN and PI3K create the ordered patterns of
cell shape is still unclear, the observed patterns have enabled us to
propose a model of spontaneous cell migration, i.e., the coordination of
migration with the formation of ordered patterns of cell shape. This
coordination is strengthened especially in WT STA cells due to spontaneous
cell polarization mediated by asymmetric PTEN localization. PTEN is
asymmetrically localized in WT STA cells, while PTEN is uniformly
distributed at the cell membrane in WT VEG cells. The asymmetric
localization of PTEN is not necessary for creating ordered patterns but is
required for persistently enforcing the direction of movement along with the
orientation of cell shape.

Elongating WT STA cells displayed the long-lasting correlation in iaCF, while
rotating or oscillating ones showed the rapid decrease of correlation in
iaCF. This result suggests that each local pseudopodia extension mediated by
PI3K dependent F-actin polymerization enables a cell not only to move
forward but also to turn to the left or right. Spontaneous cell migration in
polarized cells thereby results from mutual coordinating behavior between
ordered patterns and cell movement. We further asked a question regarding
the temporal hierarchy between the ordered pattern and cell movement: Does a
cell elongate in the direction of cell movement or move in the direction of
protrusions? Or do these two processes coordinate instantly? In order to
infer this temporal hierarchy, we examined the point at which iaCF shows the
maximum value. Given that a cell elongates in the direction of motion with a
time delay of τ after cell movement, iaCF exhibits the maximum
of correlation at
Δt = τ.
We found that iaCF showed the maximum correlation at
Δt = τ = 0
(Fig. 6E). This
observation indicates that the coordination between cell shape and cell
movement occurs instantly.

Relationship between the ordered patterns and chemotactic behavior

Our results show that ordered patterns of cell shape are mediated by PI3K and
PTEN. PI3K and PTEN are also essential for cell polarity during chemotaxis.
The responsibility of the two proteins raises a question as to what
relations exist between the ordered patterns and cell polarity in
chemotaxis. During chemotaxis, directional sensing by CRAC occurs through
the activations of G-protein coupled receptors, and then the localizations
of PI3K and PTEN are induced by extracellular chemotactic stimuli. Guidance
signals are subject to amplification and feedback regulation of cell
polarity, probably to permit the detection of small differences in a shallow
gradient. In the case of spontaneous cell migration, Sasaki et al. have
found that PI3K and Ras are autonomously activated in the absence of
chemoattractants at the sites where F-actin projections emerge, which
suggests that spontaneous cell polarity would result from the stochastic
activation of the guidance pathway G-protein-independent guidance pathways
[12].

We observed that PTEN was localized spontaneously at the rear edge in WT STA
cells, and PI3K-dependent F-actin polymerization occurred at both the
leading edge and at the rear edge. The elongation pattern indicated that the
cells amplify shallow gradients of spontaneously produced PIP3
and maintain cell polarity through PI3K and PTEN. Moreover, the switching of
ordered patterns suggests that cells stochastically choose one of the three
ordered patterns and change it to another pattern in order to ensure overall
random exploration during spontaneous cell migration. In contrast,
chemotactic cells dominantly select the elongation pattern due to signals
from G-proteins and then move toward the source of the chemoattractant by
coordinating cell movement with the selected elongation pattern. Hence, cell
polarity in chemotaxis could be considered as the property that cells
dominantly choose an elongation pattern.

Recent studies have shown that mutants lacking PI3K and/or PTEN do not
sustain a polarized cell shape but can exhibit precise chemotaxis [14], [15].
Although cell polarity that is sustained by PI3K and PTEN in chemotaxis
could be related to the elongation pattern of cell shape, neither the cell
polarity nor the elongation pattern may be necessary for chemotaxis.

Biological function of the transition of ordered patterns of cell shape

We observed that the ordered patterns of cell shape stochastically transit
from one pattern to another. This stochastic transition could add randomness
to the motion of Dictyostelium. In terms of the advantage
of randomness, we note similarities between the stochastic change of
flagella rotations of bacteria and the stochastic transition of ordered
patterns of Dictyostelium cells. Motile bacteria such as
Escherichia coli (E.coli) have
flagella that are rotating structures driven by the proton motive force.
Although Dictyostelium and E coli do not
use identical locomotory apparatus, both forage with the aid of
stochastically switching between two characteristic states, a fairly
straight motion and a turning motion. For the fairly straight motion,
bacteria rotate flagella in a clock-wise manner, and
Dictyostelium cells maintain the elongation pattern of cell
shape for at least 10 min. Moreover, for the turning motion, bacteria tumble
by counter-clockwise rotation of flagella, and
Dictyostelium cells change into the rotation or oscillation
pattern that lasts the order of 10–20 min. Stochastic switching
of these ordered patterns allows cells to do overall random exploration.

Regarding the efficiency of searching behavior, Li et al. have conducted the
numerical simulation of the amoeboid motion in which a cell finely directs
the motion with a persistent time of approximately 9 min [26]. By comparing to the random walk
(defined as the motion whose direction and speed are randomly chosen), they
found that the fairly straight motion of 9 min in the amoeboid motion allows
cells to cover more area in a given time. This non-random motion results in
the 1.6 to 2.4 fold increase of the efficiency greater than random walk. In
our study, the coordination of the ordered patterns with cell movement
suggests that the ordered patterns of cell shape attribute to the non-random
motion in foraging behavior, e.g., the straight motion directed by the
elongation pattern could realize the fairly straight motion. Based on the
discussion done by Li et al, such a non-random (persistent) motion directed
by the ordered pattern could improve the efficiency of searching behavior.

In this study, we found the ordered patterns of cell shape were mediated by
PI3K/PTEN/F-actin, and the coordination of these patterns with cell movement
constituted the mechanism of spontaneous cell migration. Wave-like
protrusions have been observed in various cells from flies to mammals [13], [27]–[29]. We
believe that the ordered patterns mediated by PI3K/PTEN/F-actin may occur in
other cells and that the application of the present approach offers the
opportunity to complement the analysis of processes such as leukocyte
migration [30]–[32].

Materials and Methods

Strains and Cultures

Wild-type, pten−, PTEN-GFP, CRAC-GFP,
ABD-GFP of AX-2, and pi3k1/2− of
AX-3 Dictyostelium discoideum were cultivated in liquid
nutrient medium (HL-5 medium) with appropriate antibiotics and harvested
axenically during exponential growth. Cells are at vegetative (VEG) state,
which is termed as VEG cells, in the rich HL-5 medium. After 6 to 7 hour of
nutrient depletion, VEG cells differentiate into the different developmental
stages, starved state. We name the cells at the STA state as starved (STA)
cells. We prepared VEG and STA cells used for our measurements as follows:
VEG cells were suspended in 10 ml phosphate buffer (PB) at a density of
103 cells/ml, and 1.5 ml of the suspension was placed onto a
1.5% agar plate (66 mm diameter, Falcon). The agar plate was
incubated for 30 min to allow the cells to adhere to the agar surface
(Nakarai). STA cells were prepared the following conventional protocol. VEG
cells were washed 3 times with PB and then plated without nutrient at a
density of 5×103 cells/cm2 for
6–7 h at 22 C°, and then washed twice by PB. All the
experiments were performed at room temperature (22 C°).

We added the PI3K inhibitor LY294002 (Sigma) to a final concentration of 100
µM 30 min before the real-time recording. No differences were
observed if the drug was added either 30 min or 60 min before observation.
Cells treated with LY294002 and the
pi3k1/2− mutants tend to detach
from the substrate during the extension of pseudopodia. In order to prevent
detachment, we placed a 1.0% agarose sheet (TaKaRa) onto
PI3K-inhibited cells.

Digital image processing

The experimental microscope setup consisted of a Leica microscope equipped
with a cooled CCD camera (ORCA-AG, Hamamatsu Photonics) with
a×20 objective. Time lapse-images were captured with a time
interval of 1 sec. We used the phase contrast equipment in conjunction with
Metamorph software (Universal Imaging Corporation). The shutter speed of CCD
camera was 31 msec. We first flattened the background noise using Metamorph,
and then a two-component Gaussian mixture model was fitted to the pixel
intensity distribution of each image using the Expectation Maximation
algorithm. From this model, a time-dependent intensity threshold was
calculated.

Circular mapping

From each binarized image, we define the position of the centroid as . In order to calculate the dynamics of the cell shape in a
2D Cartesian system, we use circular mapping: the radial amplitude of
extensions from the centroid to the cell edge are calculated for each image.
We define this length value as
Amp(θ,t). Further we define
velocity as the displacement of the centroid, at time interval 15 s. We also require 2 additional rules
to avoid calculation difficulties. First, if the function of cell contour
becomes a multivalued function, we employed the value furthest from the
centroid. Second, if we obtain an image in which the centroid does not
locate within a cell, we do not calculate any values but just adopt the
previous values. Since we obtained the ordered patterns when we adopted arc
length instead of angle θ, these rules did not
affect our conclusions.

Actin-binding domain fused GFP (ABD-GFP) was expressed for the observation of
F-actin assembly and localization in AX-2 WT cells [25]. We
performed fluorescence and phase-contrast measurements every 6 s. There was
a 2 s delay between fluorescent measurements and phase-contrast image
capture. At each time point, the cell position was identified on the
phase-contrast images, and calculated as
Amp(θ,t). We then analyzed
the fluorescence intensity data
I(r,θ,t) where
r represents the length from the centroid. Because F-actin
is accumulated nearby cell periphery not but just on the edge, measuring
I(r,θ,t) along cell
membrane did not work well. Instead, we employed the variable
Act(θ,t), which is defined
as . The variable
Act(θ,t) is the weighted
F-actin concentration by the distance from centroid r along
the radius in the direction θ and at time
t, normalized by the sum of r.
Act(θ,t) is the reliable
measure of F-actin accumulation (Fig. S6).

Correlation analysis

We define the auto-correlation functions of
Amp(θ,t) and
Act(θ,t), and the
cross-correlation function between
Act(θ,t) and
Amp(θ,t) as

where and . Averages over time and over angles are denoted
〈 〉θ,t. The range of calculation is the entire period of measurement in
time and from 0° to 360° in space.

Further we investigated the relation between velocity and morphological dynamics
Amp(θ,t) with the
identical angle cross-correlation function (iaCF) that is defined as

where . We obtained the same results on iaCF without the gaussian
filtering of velocity field.

In order to evaluate the coordination between cell shape and movement, we
calculated the probability of the deviation of
Amp(θV(t),t)
from the mean 〈Amp(θ,t)〉θ described as , where
θV(t) is the
direction of cell movement at an interval of 15 s.

We performed statistical analysis using two-tailed Student's t- test or an
analysis of variance (ANOVA). All data found to be significant by ANOVA were
compared using Tukey-Kramer test.

Spectrum analysis of cell shape

In addition to cross-correlation function between
Amp(θ,t) and
Act(θ,t), we further
examined the specialized role of PI3K and PTEN in the control of cell shape
by using power spectrum of
Amp(θ,t). We calculated the
power spectrum of Amp(θ,t) by using

where F(k,ω) is
the 2-dimensional fourier transform of
Amp(θ,t), and
F*(k,ω)
is the conjugate of F(k,ω).
k describes the spatial frequency of morphological
dynamics and ω describes temporal frequency.
This power spectrum allows us to reveal the long-range correlation of
dynamic remodeling of cell shape.

Clustering analysis

We used clustering method in order to dissect the observed patterns reliably.
We can qualitatively dissect elongation, rotation, and oscillation patterns
into the combination of the left-hand and the right-hand waves rotating
around a circle. For rotating waves, we describe:

where m is the mode number,
ω is the rate of rotation,
Q is the centrifugal amplitude. Oscillation is described by
superimposition of left-handed and right-handed rotation functions of
Qrot(Δθ,Δt)

Both functions
Qrot(Δθ,Δt)
and
Qosc(Δθ,Δt)
may be combined to give:

which describes oscillation as waves rotating around a
circle. Here, we approximate an obtained ACF,
CAmp,Amp(Δθ,Δt),
by the following function:

where m1,
m2, m3 are the mode
numbers, ω1 and
ωR are the rates of rotation
respectively, and R, QL,
QR represent the centrifugal amplitude.
R(Δθ,Δt)
is the coherent component of the vector length from centroid to a point on
the basal circle, which is describing elongation pattern. The superposition
of three functions results in a qualitative description of the patterns in
ACF.

We can obtain both the rates of rotation and the wave amplitudes of
R(Δθ,Δt),
QL-rot(Δθ,Δt),
and
QR-rot(Δθ,Δt)
by two-dimensional Fourier transform of
CAmp,Amp(Δθ,Δt).
The amplitude of each Fourier coefficient represents how large a given
spatiotemporal wave contributes to the overall dynamics of cell shape. We
note that the amplitudes of the spatial harmonics would differ depending on
the size of coordinates. In fact, the patterns of ACF are the
superimpositions of various waves with the inherent rate of rotation. For
simplicity, we focus on and choose the mode with the largest wave amplitude
among standing waves, the one among left-handed waves, and the one among
right-handed waves, respectively. Hence, we obtain the parameters of the
wave amplitudes
AS = max(R),
AL = max(QL)
and
AR = max(QR).
Hierarchical clustering was performed on the subsets of 53 and 53 cells in
WT VEG and WT STA cells, respectively. We use three parameters
AS, AL, and
AR for the clustering. However, the flip
horizontal of
CAmp,Amp(Δθ,Δt)
sometimes affects the result of clustering. To avoid this artificial
problem, we define AM and
AM by sorting AL and
AR as
AM = max(AL,AR)
and
AM = min(AL,AR).
This treatment allows us to obtain the robust clusters (Fig.
S7). Clustering was done by means of a standard single-linkage
algorithm with a Euclidean metric (Matlab, Mathworks).

Persistence of directional movement

We defined persistency of moving direction as , where s represents the distance of the
trajectory from the starting point
(t = 0), and represents the tangent vector at
s = s′.
We also defined persistence length lC as
CPL(lC) = e−1.
Persistence length is a preferable measure for determining for how far a
cell typically moves straight in a given direction.

Supporting Information

Text S1

Figure S1

Long-term measurement of the morphological dynamics of cell shape.We
measured a single WT vegetative cell for 3.3 h and then calculated the
autocorrelation function of Amp(θ, t) at each time window (500 s). Six examples
of autocorrelation function are shown on the left side of
Amp(θ,
t). We found that the ordered pattern dynamically changes; for
instance, from rotation to oscillation.

Figure S3

PI3K inactivation reduces the amplitude of pseudopodia. Average power
spectra of cell morphology. Upper: WT cells (red solid line),
WT+LY294002 cells (green dash line) and
pi3k1/2&minus cells (green
solid line). Lower: pten&minus
cells (red) and
pten&minus+LY294002
cells (green). Left is VEG state and right is STA state. The individual
power spectra of either WT cells or
pten&minus cells (pale red)
and those of PI3K-inhibited cells (pale green) were plotted. All
averaged power spectra were well fitted by
P(k)&propk−1.7
(black dash line).

Figure S4

Characterization of the centre of mass displacements. (A) Average mean
square displacement along the trajectory as a function of time and (B)
autocorrelation function of instantaneous velocity for WT (red),
pten&minus (blue) and
pi3k1/2&minus (green) in
both the VEG and STA states. All curves of the MSD fit a decaying
power-law. We adopt time interval of 1 s (for MSD) and 5 s (for the
autocorrelation function of velocity) in calculating the center of mass
displacements, respectively.

Figure S6

Act(&theta, t) : a reliable
measure of F-actin accumulation. (A) We employed
Act(&theta, t) instead
of the measure of I(r, &theta,
t) along cell membrane because F-actin accumulates nearby cell
membrane but not on the edge of cell membrane (see Figure 5A). To test the reliability
of Act(&theta, t), we
compared it with the largest intensity of F-actin along a radius from
centroid, Imax(&theta, t).
Act(&theta, t) is
proportional to Imax(&theta,
t). (B) A example of
Amp(&theta, t)
(oscillation pattern). (C)
Act(&theta, t) (left)
and Imax(&theta, t)
(right) of the example presented in (B). (D) Cross-correlation functions
(CCF). (left) CCF between
Act(&theta, t) and
Amp(&theta, t),
(right) CCF between Act(&theta,
t) and Imax(&theta,
t). The similarity of CCFs indicate that
Act(&theta, t) is a
reliable measure of F-actin accumulation as well as
Imax(&theta, t).

Figure S7

Clustering analysis of autocorrelation function. We first subject
autocorrelation function (ACF) to Fourier transform to obtain the three
parameters, AS, AL, and
AR. We then conduct clustering analysis of ACF based on
the obtained parameters. We show the clustering tree of wild-type
vegetative cells.